Optimal. Leaf size=22 \[ \sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.038355, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2194, 4844, 2246, 32} \[ \sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2194
Rule 4844
Rule 2246
Rule 32
Rubi steps
\begin{align*} \int e^x \sin ^{-1}\left (e^x\right ) \, dx &=e^x \sin ^{-1}\left (e^x\right )-\int \frac{e^{2 x}}{\sqrt{1-e^{2 x}}} \, dx\\ &=e^x \sin ^{-1}\left (e^x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x}} \, dx,x,e^{2 x}\right )\\ &=\sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0082238, size = 22, normalized size = 1. \[ \sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 18, normalized size = 0.8 \begin{align*}{{\rm e}^{x}}\arcsin \left ({{\rm e}^{x}} \right ) +\sqrt{- \left ({{\rm e}^{x}} \right ) ^{2}+1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4467, size = 23, normalized size = 1.05 \begin{align*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt{-e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00802, size = 51, normalized size = 2.32 \begin{align*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt{-e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.829462, size = 17, normalized size = 0.77 \begin{align*} \sqrt{1 - e^{2 x}} + e^{x} \operatorname{asin}{\left (e^{x} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15127, size = 23, normalized size = 1.05 \begin{align*} \arcsin \left (e^{x}\right ) e^{x} + \sqrt{-e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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